continuous rolling having occurred between the two centroids. 1 ^
These centroids we have now found completely. They are (a) for
the equilateral triangle an equilateral curve-triangle inscribed
within it, (b) for the duangle a similar duangle, which has the
minor axis of the first for its major axis, and which rolls in the
centroid of the triangle.

### §23.

Point-paths of the Duangle relatively to the
Equilateral Triangle.

(Plates I and II.)

The paths described by points in the duangle relatively to the triangle can now be completely determined; for we know the centroids of both figures, and can fix that of the triangle and set that of the duangle in motion upon it. As these paths are formed by the rolling of one centroid upon another, they all belong to the class of curves known as roulettes. We have already deter- mined the paths of two important points of the duangle, the points P and Q. These points always belong to the smaller Cardan ic circle, and so describe always parts of hypocycloids coinciding with portions of the diameter of the larger circle. These portions form, as has been already noticed, two coincident equilateral triangles, U T Q, Fig. 1 Plate I. All other points of the duangle describe necessarily arcs of prolate or curtate hypocycloids, which are, as we have mentioned, ellipses. All these prolate and curtate curves are known by the common name of trochoids. 13 "We may " therefore say that all the remaining points in the duangle have for their paths hypotrochoids, of which the equilateral triangle UTQis the foundation. As this triangle consists of six portions of hypocy- cloids, so all the other point-paths must consist of six hypotro- choidal arcs. The figures thus built up take very various forms with different positions of the describing point. Fig. 1 shows three of them external to the triangle. The describing points themselves lie upon the production of the minor axis Q P of the duangle, that is, of the major axis of the centroid Q m v Pm 2 , and are numbered 1, 2, 3, commencing with the outermost point 4, which